Dynamic polarizabilities and magic wavelengths of Sr + for focused vortex light
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Dynamic polarizabilities and magic wavelengths of Sr + for focusedvortex light Anal Bhowmik ∗ and Sonjoy Majumder † Department of Physics, Indian Institute ofTechnology Kharagpur, Kharagpur-721302, India. (Dated: October 9, 2018)
Abstract
A theory of dynamic polarizability for trapping relevant states of Sr + is presented here whenthe ions interact with a focused optical vortex. The coupling between the orbital and spin angularmomentum of the optical vortex varies with focusing angle of the beam and is studied in the cal-culation of the magic wavelengths for 5 s / → d / , / transitions of Sr + . The initial state of ourinterest here is 5 s / with m J = − / + [Phys. Rev. A , 022511 (2018)]. We find variation in magic wavelengths andthe corresponding polarizabilities with different combinations of orbital and spin angular momen-tum of the vortex beam. The variation is very significant when the wavelengths of the beam are inthe infrared region of electromagnetic spectrum. The calculated magic wavelengths will help theexperimentalists to trap the ion for performing the high precision spectroscopic measurements. ∗ [email protected] † [email protected] . INTRODUCTION Optical trapping of atoms or ions has been extensively used in high precision spectroscopicmeasurements [1, 2]. But the mechanism of trapping using a laser light inevitably producesa shift in the energy levels of the atoms involved in absorption. The shift is called thestark shift. In general, the shift is different for these energy states of the atom. Thusnaturally it will influence the fidelity of the precision measurement experiments due to non-achieving of exact resonance. However, this drawback can be diminished if the atoms aretrapped at magic wavelengths of the laser beam, for which the differential ac stark shift ofan atomic transition effectively vanishes. Therefore, the magic wavelengths have significantapplications in atomic clocks [3–5], atomic magnetometers [6] and atomic interferometers[7].All the previous studies of magic wavelengths for trapping of different atoms or ionsare obtained for the Gaussian modes of a laser [8–11]. In this work, we determine themagic wavelengths of the transitions 5 s / , − / → d / ,m J and 5 s / , − / → d / ,m J ofSr + ion, assuming the external light field is a circularly polarized focused optical vortexsuch as Laguerre-Gaussian (LG) beam [12]. Since the stark shifts will be different forthe states 5 s / , − / and 5 s / , +1 / , different laser frequency (magic frequency) should beapplied to minimize the systematic errors in the experiments involved the state 5 s / , − / compare to 5 s / , +1 / state. Therefore, it is important to quantify the magic wavelengths ofthe transitions 5 s / , − / → d / ,m J and 5 s / , − / → d / ,m J of Sr + , as we have alreadyreported the magic wavelengths related to 5 s / , +1 / state [13]. However, the special propertyof optical vortex is that, apart from the polarization (i.e., spin angular momentum (SAM)),the optical vortex carries orbital angular momentum (OAM) due to its helical phase front[14]. Now, it is well known that during the interaction of a paraxial LG beam with atoms orions (which are below its recoil limit), the quadrupole transition is the lowest-order transitionwhere the OAM of the LG beam affects the electronic motion [14, 15]. Therefore, the OAMof a paraxial LG beam does not influence dipole polarizability of an atomic state. Hence, incase of paraxial LG beam, the dipole polarizability and the magic wavelengths solely dependon the SAM of the beam. But unlike the paraxial LG beam, the OAM and SAM of theoptical vortex get coupled when the beam is focused [12]. This leads to the transfer of OAMto the electronic motion of the atoms in the dipole transition level and creates an impact2n the polarizability of an atomic state [12]. Further, the coupling of angular momentaincreases with the focusing angle. However, in this work, we quantify all these effects ofOAM and SAM on the polarizability of an atomic state regarding magic wavelengths. II. THEORY
If an atom or ion placed in an external oscillating electric field E ( ω ), then the second-order shift in a particular energy level of the atom or ion is proportional to the square of theelectric field, E ( ω ). The proportional coefficient is called the dynamic polarizability α ( ω )of the atomic or ionic energy state at frequency ω of the external electric field and it can bewritten as [16] α ( ω ) = α c ( ω ) + α vc ( ω ) + α v ( ω ) . (1)Where α c ( ω ) and α v ( ω ) are dynamic core polarizability of the ionic core and dynamicvalence polarizability of the single valence system, respectively. This ionic core is obtainedby removing the valence electron from the system. α vc ( ω ) is the correction [17] in corepolarizability in the presence of the valence electron. As the core electrons are tightlybound to the nucleus, the presence of a valence electron is expected not to change thecore polarizability significantly. Thus we consider α vc in the present method of calculationswithout variation of ω . α v ( ω ) is calculated using the external electric field of focused LGbeam [12]. In case of focused LG beam, OAM and SAM are no longer separately a goodquantum number as they get coupled to each other. Therefore, the effect of total angularmomentum (OAM+SAM) can be seen on α v ( ω ), which can be expressed as [13] α v ( ω ) = 2 A α v ( ω ) + 2 × (cid:18) m J J v (cid:19) A α v ( ω ) + 2 × (cid:18) m J − J v ( J v + 1)2 J v (2 J v − (cid:19) A α v ( ω ) , (2)where J v is the total angular momentum of the state ψ v and m J is its magnetic component.The coefficients A i s are A = h { I ( l )0 } + { I ( l ) ± } + 2 { I ( l ) ± } i , A = h ±{ I ( l )0 } ∓ { I ( l ) ± } i and A = h { I ( l )0 } + { I ( l ) ± } − { I ( l ) ± } i . The parameter I ( l ) m , where m takes the values 0, ± ±
2, depends on focusing angle ( θ max ) by [12, 18] I ( l ) m ( r ′⊥ , z ′ ) = Z θ max dθ √ r ′⊥ w sin θ ! | l | (sin θ ) | l | +1 √ cos θg | m | ( θ ) J l + m ( kr ′⊥ sin θ ) e ikz ′ cos θ . (3)3ere r ′⊥ is the projection of r ′ on the xy plane, w is the waist of the paraxial circularlypolarized LG beam which is focused by a high numerical aperture. The angular functionsare g ( θ ) = 1 + cos θ , g ( θ ) = sin θ and g ( θ ) = 1 − cos θ . α v ( ω ), α v ( ω ) and α v ( ω ) introducedin Eq.( 2) are the scalar, vector and tensor parts respectively, of the valence polarizabilityand are expressed as [16, 19] α v ( ω ) = 23(2 J v + 1) X n |h ψ v || d || ψ n i| × ( ǫ n − ǫ v )( ǫ n − ǫ v ) − ω , (4) α v ( ω ) = − s J v ( J v + 1)(2 J v + 1) X n ( − J n + J v J v J v J n |h ψ v || d || ψ n i| × ω ( ǫ n − ǫ v ) − ω , (5)and α v ( ω ) = 4 s J v (2 J v − J v + 1)(2 J v + 1)(2 J v + 3) X n ( − J n + J v J v J n J v |h ψ v || d || ψ n i| × ( ǫ n − ǫ v )( ǫ n − ǫ v ) − ω . (6)Henceforth, whenever we mention about SAM or OAM in the following text, it is consideredto be the angular momentum of the paraxial LG beam before passing through the focusinglens. III. NUMERICAL RESULTS AND DISCUSSIONS
The aim of this work is to calculate the dynamic polarizabilities of the 5 s / , 4 d / , and4 d / states for different magnetic sublevels of Sr + . The scalar, vector and tensor partsof the valence polarizabilities are calculated using Eqs (4), (5), and (6). The preciseestimations of these three parts of the valence polarizability depend on the accuracy of theunperturbed energy levels and the dipole matrices among them. In order to evaluate theseproperties, we use correlation exhaustive relativistic coupled cluster (RCC) theory [20–24]with wave operators associated with single and double and partial triple excitations in linearand non-linear forms. The wavefunctions calculated by the RCC method can produce highlyprecise E1 transition amplitudes as discussed in our recent work [13]. Calculation in thisreference yields that the static core polarizability ( α c (0)) of the ion is 6.103 a.u., and thestatic core-valence parts of the polarizabilities ( α vc (0)) for the states 5 s , 4 d and 4 d are − .
25 a.u., − .
38 a.u. and − .
42 a.u., respectively.4n order to determine the precise values of dynamic valence polarizabilities, we requirecalculating a large number of dipole matrix elements. Another way to say, the runningindex n in Eqs (4) to (6) is turning out to be around 25 for Sr + to obtain accurate valencepolarizability. Since the RCC method is computationally very expensive, we break ourtotal calculations of valence polarizability in three parts depending on their significance inthe sums of Eqs (4)–(6). The first part includes the most important contributing termsto the valence polarizabilities which involves the E P to 8 P and 4 F to 6 F . Therefore, these matrix elements arecalculated using the correlation exhaustive RCC method. The second part consists of thecomparatively less significant terms associated with E P to 12 P and 7 F to 12 F . Thus wecalculate the second part using second-order relativistic many-body perturbation theory[25]. The last part, whose contributions are comparatively further small to the valencepolarizability, includes the intermediate states from n = 13 to 25, are computed using theDirac Fock wavefunctions.In FIG. 1 and 2, we present the variations of total polarizabilities of 5 s / , − / , 4 d / ,m J and 4 d / ,m J (for different magnetic quantum number, m j , of the states) states with thefrequency of the external field of the focused LG beam. The focusing angle of the LG beamis considered 50 ◦ in both the figures. The combinations of angular momenta of the paraxialLG beam have chosen as (OAM, SAM) = (+1 , +1) and (+1 , −
1) in Fig. 1 and Fig. 2,respectively. The resonances occur in the plots due to the 5 s / → p / , / transitions for5 s / state, 4 d / → p / , / transitions for 4 d / state and 4 d / → p / transitions for4 d / state. The plots show a number of intersections between the polarizabilities of 5 s at m J = − / d , states. These intersections indicate magicwavelengths, at which the difference in the stark shifts of the two related states vanishes.Figures show that magic wavelengths which fall in the infrared region of the electromagneticspectrum have large polarizabilities compared to the magic wavelengths of the visible orultraviolet region. These magic wavelengths with high polarizabilities will be more effectiveto trap the ion, and thus they are highly recommended for trapping. These two figures aregiven as an example. Similar plots are studied for different focusing angles, say 60 ◦ and 70 ◦ ,and corresponding magic wavelengths are discussed later in this paper.In Table I, II and III, we have listed a large number of magic wavelengths along with5heir corresponding polarizabilities, when the focusing angles of LG beam are 50 ◦ , 60 ◦ and70 ◦ . The table I is for the transition 5 s / → d / , and the combinations of OAM andSAM are (+1 , +1), (+1 , − , +1) and (+2 , − s / → d / but the combinations of OAM and SAM are ((+1 , +1), (+1 , − , +1), (+2 , − m J value of 5 s / is considered − / m J = 1 /
2. There are five sets of magic wavelengthsobtained for each of the multiplets of 4 d / for all the combinations of angular momentaand focusing angles of the LG beam in the given frequency range. Whereas, in the samerange of wavelength spectrum for 4 d / state, our calculations show seven sets of magicwavelengths (see Table II and III) for most of the multiplets. Since the resonance transitionof 5 s → d , d are 687 nm and 674 nm, respectively, thus the ion is attracted and trappedto the high intensity (low intensity) region of the LG beam when the magic wavelength islarger (smaller) than the resonance wavelength.Since this work is about the finding of suitable magic wavelengths for trapping, we onlygive an estimation of the theoretical uncertainty in the calculated magic wavelengths. Herewe collect the most important set of E s → p , tran-sitions for 5 s state; 4 d → p , and 4 d → f transitions for 4 d state; 4 d → p and 4 d → f , transitions for 4 d state. We compare our RCC results with the SDpTvalues calculated by Safronova [26] and further apply those E ± IV. CONCLUSIONS
In conclusions, we find a wide list of magic wavelengths for the transitions 5 s / ( m J = − / → d / ( m J ) and 5 s / ( − / → d / ( m J ) of the Sr + ion. We have found herea quite distinct set of values of magic wavelengths compared to the same with the initialstate 5 s / ( m J = 1 / P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d3(+1/2) 4d3(-1/2) 4d3(+3/2) 4d3(-3/2) (a) P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d5(+1/2) 4d5(-1/2) 4d5(+3/2) 4d5(-3/2) 4d5(+5/2) 4d5(-5/2) (b) P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d3(+1/2) 4d3(-1/2) 4d3(+3/2) 4d3(-3/2) (c) P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d5(+1/2) 4d5(-1/2) 4d5(+3/2) 4d5(-3/2) 4d5(+5/2) 4d5(-5/2) (d)
FIG. 1. Variation of polarizabilities (Polar) of 5 s and 4 d , states with frequency (Freq) areplotted when the focusing angle of LG beam is 50 ◦ with OAM=+1 and SAM=+1. The bracketsindicate the magnitudes of different magnetic components. Fig. (a) and (c) are for the 5 s and4 d states, and Fig. (b) and (d) are for the 5 s and 4 d states.[1] C. Champenois, M. Houssin, C. Lisowski, M. Knoop, G. Hagel, M. Vedel, and F. Vedel, Phys.Lett. A
298 (2004). .00 0.02 0.04 0.06 0.08 0.10050100150200 OAM=+1, SAM=-1 P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d3(+1/2) 4d3(-1/2) 4d3(+3/2) 4d3(-3/2) (a) P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d5(+1/2) 4d5(-1/2) 4d5(+3/2) 4d5(-3/2) 4d5(+5/2) 4d5(-5/2) (b) P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d3(+1/2) 4d3(-1/2) 4d3(+3/2) 4d3(-3/2) (c) P o l a r ( a . u . ) Freq (a.u.) 5s(-1/2) 4d5(+1/2) 4d5(-1/2) 4d5(+3/2) 4d5(-3/2) 4d5(+5/2) 4d5(-5/2) (d)
FIG. 2. Variation of polarizabilities (Polar) of 5 s and 4 d , states with frequency (Freq) areplotted when the focusing angle of LG beam is 50 ◦ with OAM=+1 and SAM= −
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State (4 d / ( m J )) λ ◦ magic α λ ◦ magic α λ ◦ magic α State (4 d / ( m J )) λ ◦ magic α λ ◦ magic α λ ◦ magic α OAM=+1, SAM=+1 OAM=+1, SAM=-1 (+1 /
2) 2680.20 100.36 2201.13 104.81 1963.94 109.41 (+1 /
2) 2169.68 117.01 2080.52 118.77 2016.08 120.321069.56 114.82 1054.71 119.84 1047.43 124.19 1037.89 132.76 1023.90 134.76 1021.60 136.67422.27 20.89 421.10 20.50 417.25 20.13 404.65 7.19 404.65 7.69 404.65 8.47213.61 -26.26 214.31 -27.87 213.51 -28.82 213.21 -32.25 213.11 -32.87 214.01 -33.72198.27 -20.34 198.45 -21.32 198.71 -22.37 200.72 -26.41 200.72 -26.69 200.81 -27.23( − /
2) 8136.31 98.17 4339.37 102.62 3120.78 106.42 ( − /
2) 1693.80 119.80 1668.99 121.56 1650.85 122.761069.56 114.98 1042.64 119.56 1040.26 124.45 1021.60 133.40 1019.31 135.28 1014.77 137.07421.88 -0.84 420.71 0.33 416.87 2.29 404.65 20.87 404.65 20.76 404.65 20.22213.11 -26.13 214.11 -27.80 213.21 -28.51 213.21 -32.25 213.11 -32.87 214.31 -33.80200.19 -20.96 200.28 -22.04 200.37 -23.13 199.31 -25.72 199.40 -26.33 199.49 -26.61(+3 /
2) 1074.61 114.75 1079.70 117.94 1082.26 122.18 (+3 /
2) 1875.04 118.41 1875.04 119.83 1875.04 121.14911.27 122.62 916.77 126.69 920.47 130.40 971.50 136.00 977.75 137.55 977.75 138.96422.67 46.61 421.49 44.61 417.25 42.08 404.29 -5.33 404.29 -4.11 404.29 -2.46213.91 -26.39 214.31 -27.87 213.61 -28.82 212.91 -31.91 212.91 -32.53 213.31 -33.37198.71 -20.47 198.71 -21.40 198.71 -22.51 204.78 -28.38 204.23 -28.50 203.77 -28.53( − /
2) 1759.20 103.64 1786.80 106.72 1800.92 110.43 ( − /
2) 1114.02 129.69 1130.60 130.60 1199.04 130.05957.21 120.05 957.21 124.22 963.28 127.90 935.59 138.21 939.45 139.68 945.30 141.12421.49 -17.31 420.71 -15.76 416.87 -12.98 405.01 35.93 405.01 34.62 405.01 33.22212.52 -28.35 213.21 -32.25 213.11 -32.87 214.31 -33.80208.53 -26.52 198.97 -25.49 199.05 -25.99 199.14 -26.42
OAM=+2, SAM=+1 OAM=+2, SAM=-1 (+1 /
2) 2462.88 101.83 2052.40 107.37 1822.53 113.66 (+1 /
2) 2149.21 117.80 2052.40 119.38 1955.51 121.231057.15 116.84 1052.27 122.12 1037.89 127.22 1021.60 134.04 1023.90 136.07 1019.31 137.47419.55 20.60 418.01 20.67 415.72 21.18 404.65 7.20 404.65 7.77 404.65 8.96213.21 -26.96 213.21 -27.86 213.11 -29.72 213.81 -32.84 213.31 -33.02 213.81 -33.82198.36 -20.70 1986.20 -21.98 198.88 -23.52 200.81 -26.65 200.72 -26.97 200.81 -27.44( − /
2) 6417.37 100.05 3704.34 104.54 2696.06 110.28 ( − /
2) 1687.53 119.86 1662.90 121.92 1633.10 123.971047.43 117.88 1040.26 122.15 1037.89 127.22 1021.60 134.04 1019.31 136.45 1008.04 138.16419.17 0.34 417.63 1.67 415.35 3.45 405.01 20.78 405.01 20.42 405.01 20.08213.21 -26.96 213.21 -27.86 213.11 -29.72 214.21 -33.10 213.71 -33.31 214.11 -34.02200.19 -21.41 200.37 -22.68 200.54 -24.18 199.31 -25.96 199.49 -26.46 199.58 -26.88(+3 /
2) 1077.15 116.11 1082.26 119.92 1090.03 125.50 (+3 /
2) 1875.04 118.48 1875.04 120.23 1859.73 121.61913.09 123.69 918.62 129.07 924.21 133.98 975.66 136.02 977.75 138.62 979.86 139.45419.94 45.49 418.40 43.48 415.72 39.44 404.29 -4.96 404.65 -3.42 404.65 -1.53213.21 -26.96 213.21 -27.86 213.11 -29.72 213.01 -32.49 213.01 -32.97 213.21 -33.53198.71 -20.82 198.71 -22.04 198.79 -23.48 204.60 -28.39 203.95 -28.51 203.50 -28.66( − /
2) 1772.89 105.38 1793.83 109.29 1822.53 113.66 ( − /
2) 1119.49 130.00 1139.08 131.07 1238.13 130.00955.21 122.16 961.25 125.59 967.37 131.68 937.52 138.77 941.39 140.31 947.26 141.51419.17 -16.99 417.63 -14.64 415.35 -11.16 405.01 35.64 405.01 33.98 405.01 32.12212.22 -27.86 212.71 -29.20 214.31 -33.18 213.81 -33.42 214.21 -34.17209.78 -26.52 207.48 -26.86 199.05 -25.81 199.05 -26.17 199.14 -26.63 ABLE II. Magic wavelengths (in nm) of Sr + for different focusing angles 50 ◦ , 60 ◦ and 70 ◦ of theLG beam for the transitions 5 s / ( − / → d / ( m J ). Non-paraxial LG beam
State (4 d / ( m J )) λ ◦ magic α λ ◦ magic α λ ◦ magic α State (4 d / ( m J )) λ ◦ magic α λ ◦ magic α λ ◦ magic α OAM=+1, SAM=+1 OAM=+1, SAM=-1 (+1 /
2) 5841.46 98.42 2920.73 102.94 2255.61 108.35 (+1 /
2) 1875.04 118.28 1808.07 120.28 1766.02 21.761077.15 114.82 1116.75 116.27 1105.91 121.27 1079.70 131.17 1082.26 132.32 1072.08 134.34617.39 175.83 616.55 182.48 614.06 189.51589.44 192.13 592.50 194.22 596.38 199.86419.55 27.71 418.78 27.40 417.25 26.66 404.65 6.03 404.65 6.82 404.29 7.99212.32 -25.66 212.12 -27.38 212.32 -28.26 212.32 -31.54 212.32 -32.58 212.32 -32.52202.59 -21.92 202.59 -22.97 202.68 -24.18 200.90 -26.58 200.99 -26.96 201.16 -27.34( − /
2) 14697.86 97.94 4032.16 102.43 2744.78 107.01 ( − /
2) 1766.02 119.13 1719.37 120.76 1687.53 122.431077.15 114.82 1116.75 116.27 1105.91 121.27 1079.70 130.99 1082.26 132.32 1072.08 134.34419.55 -4.56 418.40 -2.99 416.87 -0.42 404.65 25.76 404.65 25.14 404.29 25.28212.32 -25.66 212.12 -27.38 212.12 -28.26 212.32 -31.54 212.32 -32.58 212.32 -32.52199.05 -20.64 199.49 -21.48 199.75 -22.95 202.86 -27.35 202.86 -27.76 202.86 -28.18(+3 /
2) 1489.00 106.01 1484.15 109.22 1479.33 113.58 (+3 /
2) 1739.06 119.54 1719.37 120.76 1697.59 122.351095.27 113.95 1125.02 116.27 1116.75 121.17 1082.26 130.85 1087.43 132.17 1072.08 134.34631.07 170.76 631.07 176.44 630.20 182.83 668.08 177.61 664.19 180.87566.00 208.97 570.25 212.34 570.97 217.89 648.13 184.20 649.05 186.09420.33 61.77 419.17 59.39 417.25 55.85 404.65 -12.78 404.65 -11.43 404.29 -8.59212.32 -25.66 212.12 -27.38 212.12 -28.26 212.32 -31.54 212.32 -32.58 212.32 -32.52205.61 -23.12 205.43 -24.00 205.06 -25.42 198.88 -25.54 199.14 -26.08 199.40 -26.58( − /
2) 2109.41 101.90 2007.20 105.43 1890.60 110.08 ( − /
2) 1479.33 121.89 1474.54 123.15 1479.33 124.731084.84 114.38 1116.75 116.27 1114.02 121.23 1087.43 130.75 1092.65 132.01 1077.15 134.14694.56 150.71 691.40 155.63 687.23 163.29 628.46 192.56 626.73 196.34 625.01 198.77643.55 165.68 644.46 169.06 646.29 175.62 575.29 222.87 577.48 225.62 578.95 226.73419.17 -35.11 418.01 -31.33 416.48 -26.87 405.01 45.89 404.65 43.96 404.29 41.92212.32 -25.66 212.12 -27.38 212.12 -28.26 212.32 -31.54 212.32 -32.58 212.32 -32.52195.22 -19.20 195.89 -20.45 196.82 -21.76 204.50 -28.12 204.32 -28.49 204.23 -28.82(+1 /
2) (+1 /
2) 1455.70 122.30 1474.54 123.15 1493.88 124.251097.91 130.25 1097.91 131.69 1082.26 133.88635.47 168.88 633.70 176.44 633.70 181.31 697.75 169.21 694.56 171.81 690.35 175.53537.94 237.89 542.42 241.88 547.64 243.35 643.55 186.01 644.46 187.88 644.46 190.28420.33 98.74 419.55 93.34 417.63 86.15 404.65 -32.05 404.65 -28.77 403.93 -24.62212.32 -25.66 212.12 -27.38 212.12 -28.26 212.32 -31.54 212.32 -32.58 212.32 -32.52208.91 -24.48 208.34 -25.49 207.86 -26.20 196.48 -24.61 196.99 -25.12 197.67 -25.82( − /
2) 1228.12 113.97 1279.87 116.64 ( − /
2) 1262.14 125.78 1276.28 127.13 1287.10 127.991130.60 116.13 1127.81 120.74 1111.30 129.79 1111.30 131.27 1095.27 133.42786.93 134.41 760.66 140.86 740.87 151.66 631.95 190.75 631.07 193.16 631.07 196.65640.83 167.80 641.74 173.42 642.64 177.94 563.21 233.46 569.54 231.24 570.25 232.41419.17 -62.67 418.01 -58.24 416.10 -51.20 405.01 67.44 404.65 64.11 404.65 60.55212.32 -25.66 212.12 -27.38 212.12 -28.26 212.32 -31.54 212.32 -32.58 212.32 -32.52189.06 -17.17 190.56 -18.38 192.49 -20.11 206.82 -29.23 206.45 -29.52 205.98 -29.63 ABLE III. Magic wavelengths (in nm) of Sr + for different focusing angles 50 ◦ , 60 ◦ and 70 ◦ of theLG beam for the transitions 5 s / ( − / → d / ( m J ). Non-paraxial LG beam
State (4 d / ( m J )) λ ◦ magic α λ ◦ magic α λ ◦ magic α State (4 d / ( m J )) λ ◦ magic α λ ◦ magic α λ ◦ magic α OAM=+2, SAM=+1 OAM=+2, SAM=-1 (+1 /
2) 3927.88 100.17 2517.31 106.00 2052.40 112.18 (+1 /
2) 1852.17 118.93 1793.83 120.89 1739.06 122.921077.15 115.89 1103.23 119.70 1069.56 126.17 1087.43 131.29 1074.61 133.39 1079.70 135.08617.39 177.85 614.89 186.71 612.41 196.04590.96 193.35 594.05 198.04 600.31 202.71419.55 27.04 418.01 26.84 415.72 26.52 404.29 5.89 404.29 6.90 404.29 8.39212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14202.59 -22.30 202.68 -23.71 202.77 -24.98 200.90 -26.66 201.07 -27.13 201.34 -27.69( − /
2) 6328.24 99.71 3120.78 105.14 2289.62 111.36 ( − /
2) 1739.06 119.79 1700.13 121.56 1650.85 123.611077.15 116.05 1103.23 119.70 1069.56 126.17 1090.03 131.10 1077.15 133.27 1079.70 135.05419.17 -4.40 417.63 -2.19 415.72 0.68 404.29 26.20 404.29 25.68 404.29 24.43212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14199.14 -20.73 199.66 -22.34 200.10 -23.84 202.86 -27.49 202.86 -27.92 202.86 -28.32(+3 /
2) 1484.15 107.39 1479.33 111.66 1479.33 116.62 (+3 /
2) 1732.45 119.83 1706.49 121.50 1681.30 123.421095.27 115.33 1114.02 119.05 1082.26 125.77 1092.65 130.97 1077.15 133.19 1082.26 134.95631.07 172.27 630.20 179.39 629.33 188.18 667.11 179.06 660.34 183.18567.41 209.40 570.25 214.61 572.40 222.72 648.13 184.71 649.05 186.68419.94 61.30 418.40 57.72 416.10 53.64 404.29 -12.25 404.29 -9.90 404.29 -7.10212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14205.52 -23.18 205.24 -24.35 204.87 -25.69 198.97 -25.73 199.31 -26.30 199.66 -26.87( − /
2) 2071.06 103.32 1947.15 107.98 1815.27 113.42 ( − /
2) 1474.54 122.38 1479.33 123.94 1479.33 125.581087.43 115.55 1111.30 119.44 1074.61 126.06 1097.91 130.71 1082.26 132.99 1084.84 134.74694.56 152.80 689.31 159.51 681.07 169.57 627.59 193.70 626.73 196.76 625.01 200.22644.46 167.38 646.29 173.59 646.29 180.33 576.02 223.86 577.48 226.28 579.69 227.60418.78 -34.39 417.25 -29.38 415.35 -23.95 404.29 45.43 404.65 43.94 404.65 41.00212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14195.30 -19.68 196.31 -21.02 197.50 -22.74 204.41 -28.21 204.32 -28.63 204.14 -29.02(+1 /
2) 1165.30 123.12 (+1 /
2) 1460.36 122.73 1484.15 123.80 1498.79 125.281136.24 123.94 1105.91 130.59 1090.03 132.75 1090.03 134.53635.47 171.08 633.70 178.77 632.82 186.03 695.62 170.93 691.40 174.00 686.20 178.48539.21 239.06 545.02 242.09 552.28 243.70 643.55 187.15 644.46 189.10 645.37 191.67420.33 97.01 418.78 90.44 416.48 80.76 403.93 -30.40 403.93 -26.88 404.29 -22.59212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14208.72 -24.75 208.15 -26.05 207.58 -27.21 196.56 -24.70 197.33 -25.37 198.02 -26.09( − /
2) 1168.29 113.24 1262.14 115.47 1328.38 119.09 ( − /
2) 1265.65 126.00 1279.87 127.27 1294.41 128.651136.24 113.69 1127.81 118.73 1103.23 124.89 1116.75 130.07 1100.56 132.28 1100.56 134.26776.21 137.65 750.63 146.66 725.53 158.38 631.95 191.87 631.95 194.58 630.20 197.79640.83 168.57 641.74 174.84 642.64 182.21 565.30 233.57 569.54 231.96 570.97 234.22418.40 -61.56 417.25 -55.41 414.97 -46.65 404.29 66.53 404.65 62.78 404.65 57.52212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14189.45 -17.39 191.44 -19.02 193.80 -21.77 206.73 -29.36 206.17 -29.50 205.70 -29.762) 1265.65 126.00 1279.87 127.27 1294.41 128.651136.24 113.69 1127.81 118.73 1103.23 124.89 1116.75 130.07 1100.56 132.28 1100.56 134.26776.21 137.65 750.63 146.66 725.53 158.38 631.95 191.87 631.95 194.58 630.20 197.79640.83 168.57 641.74 174.84 642.64 182.21 565.30 233.57 569.54 231.96 570.97 234.22418.40 -61.56 417.25 -55.41 414.97 -46.65 404.29 66.53 404.65 62.78 404.65 57.52212.22 -26.23 212.32 -27.71 212.22 -29.44 212.22 -32.29 212.52 -32.56 212.22 -33.14189.45 -17.39 191.44 -19.02 193.80 -21.77 206.73 -29.36 206.17 -29.50 205.70 -29.76